Arguments with Quantified Statements

1
Arguments with Quantified Statements

• Section 2.4

2
Universal Instantiation

• If some property is True of everything in a
  domain, then it is True of any particular thing
  in the domain This is the fundamental tool in
  deductive reasoning
• Example
• Everything that is alive is growing
  older Universal
• John is alive Particular Instance
• ? John is growing older
• So, we combine UI with Modus Ponens to get
  Universal Modus Ponens
• Modus Ponens UI Universal Modus Ponens
• P ? Q ?x, P(x) ? Q(x)
• P P(a) for a particular a
• ? Q ? Q(a)

3
UMP Examples

• Using Universal Modus Ponens or universal
  instantiation, fill in the following
• If a and b are real numbers, then (ab)(a-b) a2
  b2
• x and y are real numbers
• ?(xy)(x-y) x2 y2
• Universal instantiation
• Any odd integer x can be written in the form x
  2k1 for some integer k
• y is an odd integer
• ?y 2k1 for some integer k
• Universal Modus Ponens

4
UMP Examples

• Using Universal Modus Ponens or universal
  instantiation, fill in the following
• All dogs wag their tails
• An Irish wolfhound is a type of dog
• ?All Irish wolfhounds wag their tails
• Universal Modus Ponens
• If an animal is at the top of the food chain,
  then it is carnivorous
• Tigers are at the top of the food chain
• ?Tigers are carnivorous
• Universal instantiation

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Universal Modus Tollens

• Example
• Everything that is alive is growing
  older Universal
• John is not growing older Particular Instance
• John is not alive
• So, we combine UI with Modus Tollens to get
  Universal Modus Tollens
• Modus Tollens UI Universal Modus Ponens
• P ? Q ?x, P(x) ? Q(x)
• Q Q(a) for a particular a
• ? P ? P(a)

6
UMT Examples

• Using Universal Modus Tollens, fill in the
  following
• People who are good at logic made good
  programmers
• Buffy would not be a good programmer
• ?Buffy is not good at logic
• My professors are happy when I pay close
  attention to their lectures
• I fell asleep during my class today
• ?My professors are not happy today

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UMP Examples

• Using Universal Modus Tollens, fill in the
  following
• Animals at the bottom of the food chain are very
  nervous
• Lions are not nervous
• ?Lions are not at the bottom of the food chain
• ?x,k?Z, if x 2k for some integer k, then x is
  even
• X is not even
• ?x ? 2k for any k

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Proving Validity of Quantified Statements

• An argument is valid iff its form is valid
• Valid by MP/UMP or MT/UMT
• Invalid by inverse or converse error
• Examples

All rich people are happy Matt is not happy ?
Matt is not rich
?x, rich(x) ? happy(x) happy(Matt) ?
rich(Matt) Valid by Modus Tollens
All rich people are not happy Jill is not rich ?
Jill is happy
?x, rich(x) ? happy(x) rich(Jill) ?
happy(Jill) Invalid by inverse error
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More on Validity of Quantified Statements
All rich people are not happy David is not
happy ? David is rich
?x, rich(x) ? happy(x) happy(David) ?
rich(David) Invalid by converse error
All rich people are happy Wayne is rich ? Wayne
is happy
?x, rich(x) ? happy(x) rich(Wayne) ?
happy(Wayne) Valid by Modus Ponens
All rich people are not happy Kris is happy ?
Kris is not rich
?x, rich(x) ? happy(x) happy(Kris) ?
rich(Kris) Valid by Modus Tollens
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Using Diagrams to Test for Validity

• Consider the following argument
• All human beings are mortal
• Zeus is not mortal
• ? Zeus is not a human being

?x, P(x) ? Q(x) Q(Zeus) ? P(Zeus) Valid by UMT

• Zeus

human
combining
Since the Zeus dot is outside the mortal disk, it
is also outside the human disk ? conclusion
follows from the premise

• Zeus

human
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Invalidity with Diagrams

• Consider the following argument
• All human beings are mortal
• Felix is mortal
• ? Felix is a human being

?x, P(x) ? Q(x) Q(Felix) ? P(Felix) Invalid by
converse
All we know is that Felix is somewhere in the
mortals disk, but where it is wrt the human disk
is unknown ?conclusion does not necessarily
follow from the premise
human

• Felix

combining
OR

• Felix

• Felix

human
human
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Other Good Stuff to Know

• Evaluating Predicates with Truth Tables
• Given the domain
• H Adam, Eve, Rosalyn, Pete, Mario
• And these propositions hold True
• male(Adam) greedy(Adam) kind(Mario)
• male(Pete) greedy(Pete) kind(Eve)
• male(Mario)
• Is this formula True?
• ?x?H, male(x) ? greedy(x) ? kind(x)

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Evaluating Predicates with Truth Tables
x male(x) greedy(x) kind(x)
greedy(x) ? kind(x) m ? g ? k
Adam Eve Rosalyn Pete Mario
T T F
T T F F
T T
T F F F
F T T
T F T
T T F T
T T
All True, therefore the truth of the statement
?x?H, male(x) ? greedy(x) ? kind(x) is
established!
However, ?x?H, greedy(x) ? kind(x) is a false
formula!
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More on Evaluating Propositions

• Let P(x,y) be the statement xy lt xy where the
  domain for x and y is the set of all positive
  integers.
• State whether each of the following propositions
  is True or False
• ____P(4,7)
• ____P(1,8)
• ____?x ?y P(x,y)
• ____?x ?y P(x,y)
• ____?x ?y P(x,y)

T
11 lt 28
9 !lt 9
F
F
9 !lt 9